False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: flux is independent of the parametrization without an orientation condition

Statement

Flux through a parametrized surface is independent of the chosen parametrization without any orientation condition.

Facts & Assumptions

Given: The horizontal unit-square parametrizations $φ (u, v) = (u, v, 0)$ and $ψ (s, t) = (t, s, 0)$ , and the constant field $F = (0, 0, 1)$ .

[L1]

The cross product has the displayed coordinate formula; injectivity and a nonzero interior parameter cross product make a regular patch; and flux is the integral of the field dotted with the induced oriented area vector (The cross product in $R^{3}$ , Regular parametrized surface patches on compact Jordan parameter regions, Unit normal fields, orientations, and flux through a regular surface patch).

[L2]

Orientation-preserving reparametrizations preserve flux and orientation-reversing ones negate it (Flux is invariant under orientation-preserving reparametrization and changes sign under reversal).

Refutation

technique · direct

1.1

By [L1], $φ_{u} \times φ_{v} = (0, 0, 1)$ and $ψ_{s} \times ψ_{t} = (0, 0, - 1)$ . Both maps are injective, so these nonzero interior cross products make them regular patches.

givenL1algebra

2.1

The two flux integrands are therefore $1$ and $- 1$ on the unit square, so the fluxes are $1$ and $- 1$ .

step 1.1algebra

3.1

The coordinate swap has determinant $- 1$ , and [L2] explains the sign change. Since the two values differ, the orientation-free statement is false.

step 1.1step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

University of Toronto MAT237 notes, Section 5.3, An Invariance Property (standard reference, not scraped)